Harmonic conjugate

A harmonic function $v$ paired with $u$ so that $u+iv$ is analytic.

Given a harmonic function $u(x,y)$ on a domain, a harmonic conjugate of $u$ is a harmonic function $v(x,y)$ such that $u$ and $v$ together satisfy the Cauchy-Riemann equations, making $f=u+iv$ analytic there. A harmonic conjugate, when one exists on the domain in question, is unique up to adding a real constant, and finding it is essentially a matter of integrating the Cauchy-Riemann equations.

Examples

$v(x,y)=2xy$ is a harmonic conjugate of $u(x,y)=x^2-y^2$ , since together they form the analytic function $f(z)=z^2$ .
Adding any constant, such as $v(x,y)=2xy+5$ , gives another valid harmonic conjugate of the same $u$ .